The Principle of Covariance and the Hamiltonian Formulation of General Relativity |
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Authors: | Massimo Tessarotto Claudio Cremaschini |
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Institution: | 1.Department of Mathematics and Geosciences, University of Trieste, Via Valerio 12, 34127 Trieste, Italy;2.Research Center for Theoretical Physics and Astrophysics, Institute of Physics, Silesian University in Opava, Bezručovo nám.13, CZ-74601 Opava, Czech Republic |
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Abstract: | The implications of the general covariance principle for the establishment of a Hamiltonian variational formulation of classical General Relativity are addressed. The analysis is performed in the framework of the Einstein-Hilbert variational theory. Preliminarily, customary Lagrangian variational principles are reviewed, pointing out the existence of a novel variational formulation in which the class of variations remains unconstrained. As a second step, the conditions of validity of the non-manifestly covariant ADM variational theory are questioned. The main result concerns the proof of its intrinsic non-Hamiltonian character and the failure of this approach in providing a symplectic structure of space-time. In contrast, it is demonstrated that a solution reconciling the physical requirements of covariance and manifest covariance of variational theory with the existence of a classical Hamiltonian structure for the gravitational field can be reached in the framework of synchronous variational principles. Both path-integral and volume-integral realizations of the Hamilton variational principle are explicitly determined and the corresponding physical interpretations are pointed out. |
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Keywords: | Einstein-Hilbert variational principle Hamiltonian theory of GR ADM Hamiltonian theory manifest covariance |
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